Abstract

We review recent progress on holographic renormalization in the context of the gauge-gravity correspondence when the bulk geometry is not asymptotically AdS. The prime example is the Klebanov-Strassler background, whose dual gauge theory has logarithmically running couplings at all energy scales. The presented formalism provides the counterterms necessary for obtaining finite two-point functions of the scalar operators in the corresponding dual gauge theories. The presentation is self-contained and reviews all the relevant background material concerning a gauge-invariant description of the fluctuations around holographic renormalization group backgrounds.

1. Introduction

Gauge/string duality offers an alternative approach to aspects of nonAbelian gauge theories that are hard to describe with conventional techniques. For example, at strong coupling, many nonAbelian gauge theories exhibit confinement, the familiar yet still somewhat mysterious phenomenon that the only finite-energy states are singlets under the color gauge group: in colliders, we never see quarks directly, only colorless hadrons. The details of confinement, and other phenomena such as chiral symmetry breaking, are difficult to capture with conventional gauge theory methods—they are inherently nonperturbative. Instead, in the dual string picture, the nonperturbative gauge theory regime is typically described by weakly coupled closed strings propagating on a space of higher dimensionality (the “bulk”), and their dynamics can be approximated by classical supergravity. In this case, one also speaks of gauge/gravity duality.

One of the most powerful applications of gauge/gravity duality is the calculation of field theory correlation functions from the dual bulk dynamics. This idea was developed in [1–3] for superconformal gauge theories, whose gravity duals are Anti-de Sitter (AdS) spaces. Let us briefly illustrate with a simple example how such a calculation is done. The starting point is the correspondence formula [3]
where, on the left-hand side, denotes the renormalized bulk on-shell action evaluated as a functional of suitably defined boundary values of the various bulk fields. The right-hand side represents the QFT generating functional for connected correlation functions of the operators , and the play the role of sources. It is often more practical to consider the exact one-point functions of the QFT operators, which are obtained from (1.1) as

Let us consider a massless scalar field in imensional AdS space, with a bulk action
where the bulk metric is taken to be
This is AdS space with characteristic length (or “radius”) . The boundary is located at . The equation of motion for is
where . In momentum space, the general solution of (1.5), which is regular in the AdS bulk space (in particular, for ), can be expressed as
where , and is a modified Bessel function. The overall constant has been chosen such that, for , one has [4]
so that is the boundary value, which plays the role of the source, , in the dual QFT.

To calculate the on-shell action, we introduce a cutoff boundary at , integrate (1.3) by parts and use the equation of motion in the bulk part, leading to
Then, substituting (1.7) and switching to momentum space yields
where the ellipses denote terms that vanish in the near-horizon limit . The two divergent terms in the near-horizon limit are removed by adding counterterms, which must take the form of local, boundary covariant functionals of the full fields (not of the boundary values ). Hence, we add
where is the metric on the cutoff boundary, and is an arbitrary constant of dimension mass. Changing is, therefore, the same as adding some finite counterterms. After putting everything together and taking the near-horizon limit, we end up with the renormalized on-shell action
where , a constant of dimension mass that depends on , represents the renormalization scale. In (1.11), one can recognize the generating functional for the CFT 2-point function of operators of dimension 4.

This short example illustrates the general philosophy behind holographic renormalization (HR) and shows that the procedure has certain features in common with the usual renormalization procedure in QFT. First, the bare, regularized generating functional diverges when removing the cutoff. Second, counterterms are added at the cutoff, after which the cutoff can be removed. Third, different renormalization schemes differ from each other by finite (and local) counterterms. Last, the necessity to introduce a dimensionful constant (renormalization scale) gives rise to a scale anomaly. For a more detailed account of HR in AdS/CFT, valid in any asymptotically locally AdS (aAdS) bulk space-time and including also gravity, fermion and form fields, we refer to the extensive original literature and reviews [5–15]. The little “a” in aAdS is particularly important, because it allows to calculate correlators in QFTs with running couplings, where the flow away from the UV fixed point is driven either by an operator insertion or a spontaneous breaking of the scale symmetry. These are known in AdS/CFT as holographic RG flows.

There are many gauge/gravity dualities, which do not involve aAdS bulk spaces. For example, a field that has attracted much interest recently is nonrelativistic AdS/CFT, because of its potential applications to condensed matter physics. Let us just note that HR has been formulated for backgrounds with Schrödinger [16, 17] and Lifshitz-like scale symmetries [18, 19]. These and other cases share the fact that their bulk backgrounds possess some anisotropic conformal infinity, which turns out to be crucial for HR [20]. Other nonconformal cases are given by with a nonvanishing dilaton. These cases exhibit a generalized conformal structure and imply couplings that run with a power law in the UV. HR has been carried out in [21, 22] following the standard method as reviewed in [12].

In this review, we will be concerned with relativistic QFTs, whose gravity duals are formally given by so-called “fake” SUGRAs [23]. These theories describe gravity coupled to an arbitrary number of dynamical scalars, with the condition that the scalar potential can be expressed via a “superpotential”. (Hence, “fake” just means that the theory is not necessarily supersymmetric, because this condition is weaker than supersymmetry. The relation between supergravity and fake supergravity was analyzed in [24, 25].) This implies the existence of BPS domain wall background solutions, which are the holographic duals of RG flows. AdS/CFT systems (with aAdS geometry) typically fall into this class of theories. What we drop, therefore, is only the requirement that the background has an asymptotic AdS region. On the QFT side of the duality, this implies to give up the existence of a UV conformal fixed point. (For this reason, we can speak of “nonAdS/nonCFT” dualities. These play a prominent role in holographic models of QCD-like theories showing features such as mesons, chiral symmetry breaking and confinement. Further details on this subject can be found in the reviews [26–28].) In contrast to the nonconformal systems mentioned earlier, in which one can perform HR based on the general asymptotic structure of the bulk space-time and fields, we will make no specific assumptions on the asymptotic geometry of the bulk. Thus, a general treatment of HR along the lines of the standard method [12] appears to be possible only on a case-by-case basis. What we would like to answer is the question whether the fake SUGRA structure of the bulk theories can be exploited to perform HR for the QFTs dual to the BPS domain wall backgrounds. The answer to this question is positive, and we review recent progress on a perturbative, or order-by-order, approach to HR. In this approach, one considers the fluctuations around the exact BPS background flow and removes the divergences order by order in the fluctuations. So far, the second-order counterterms have been constructed, which are sufficient for the calculation of two-point functions in flat backgrounds [29]. In a sense, the approach is inspired by [13, 14], where the philosophy was put forward to concentrate on the part of the counterterm action which is really necessary to calculate -point functions for a given , that is, the terms of order in the fluctuations. In this spirit, [29] considered the case . It might be possible to derive the counterterms from a fully covariant expression (similar to [30]), but this has not been achieved yet. The approach is based on the gauge-invariant formalism for the dynamics of the bulk fluctuations [31–33], which we include in this review for completeness. This formalism has turned out to be very useful for the holographic calculation of correlation functions, mass spectra and scattering amplitudes, both in aAdS and nonaAdS backgrounds.

The prime example of a nonaAdS/nonCFT fake SUGRA system, which exhibits a logarithmically warped AdS geometry in the asymptotic region, is the Klebanov-Strassler (KS) solution [34], which is well approximated in the UV by the Klebanov-Tseytlin (KT) solution [35]. The fake SUGRA systems can be obtained by a consistent truncation of type-IIB SUGRA [33, 36–38] and are the gravity duals of an gauge theory undergoing a series of Seiberg dualities, . Whereas the KT solution correctly describes the duality cascade, its bulk singularity makes it unreliable in the IR. The resolution of the singularity in the KS background provides a geometrical description of chiral symmetry breaking and confinement. Calculating correlation functions in these cases is much more involved, also because the procedure of HR has not been worked out yet in a systematic way similar to the aAdS case. As a result, only a few attempts to calculate correlators using the KT background have been made, compare for example, [30, 39–41], and only in [30, 41] the program of HR, as reviewed in [12], was applied. A generalization of this procedure to include flavor degrees of freedom was discussed in [42]. Furthermore, calculations of mass spectra in the KS background [39, 43–47] have been done using a pragmatic approach assuming that a consistent method of HR in nonaAdS backgrounds exists.

Other systems of recent interest are derived from the Maldacena-Nunez background [48], the gravity dual of SYM describing, in the UV, the exact NSVZ -function [49] and a nonzero gluino condensate in the IR that breaks the chiral symmetry. In particular, so-called walking solutions have been considered [50–52], where in a certain regime the gauge coupling changes very little—it walks as opposed to runs. The walking region imitates, in many respects, a conformal fixed point, and the slightly broken scale symmetry implies the existence of a light particle. Walking Technicolor [53] is a phenomenologically interesting alternative to the Higgs sector of the Standard Model. Glueball mass spectra in backgrounds with walking couplings have been calculated in [54, 55].

There is, therefore, continued interest in understanding better the various aspects of HR in nonaAdS backgrounds. As far as the calculation of correlation functions is concerned, the state of the art are the mass spectra that have been mentioned above. To put these calculations on a more solid footing and to prepare the ground for further investigations (fully momentum-dependent propagators, three-point functions, etc.), a consistent and systematic procedure of HR should be constructed.

Let us now state the main idea of the order-by-order approach. The starting point of the holographic calculation of correlation functions in AdS/CFT, which we generalize to nonaAdS configurations, is the correspondence formula (1.1). In AdS/CFT, the sources are identified with asymptotically rescaled boundary values of the on-shell bulk fields. When trying to generalize this identification, one has to be aware of some implicit assumptions that one usually uses in AdS/CFT [33]. The existence of a fixed point, where the bulk geometry would be global AdS, allows to parameterize the scalars in such a way that their origin is located at the fixed point, and that the individual scalars are the duals of boundary operators with certain scaling dimensions. The fixed point corresponds to some conformally invariant QFT (CFT). Therefore, holographic RG backgrounds that flow away from the fixed point can be considered either as the duals of deformations of the CFT by relevant operators or as the duals of vacua with spontaneously broken conformal symmetry, with finite couplings or vacuum expectation values (VEVs), respectively, which are related to the background scalars. In the nonaAdS case, such an interpretation is not directly possible. The absence of a fixed point does not allow to fix an origin or to choose some otherwise preferred parameterization of the scalar fields. On the field theory side, it is not possible to say that the RG flow is due to some operator deformation or a ground state with a VEV of some special QFT. All we seem to be able to say is that the holographic RG background is the dual of some QFT that has running couplings. (In AdS/CFT, one would simply say that the QFT under consideration is the CFT + deformations or in a spontaneously broken phase.) Just as in AdS/CFT, however, correlation functions of this QFT can be obtained by studying the dynamics of the fluctuations of the bulk fields around the holographic RG flow. For two-point functions, which we will be concerned with, it is sufficient to solve the linearized equations of motion, which have a useful form in the gauge-invariant formalism. The sources will be given by certain asymptotic boundary values of the on-shell bulk fluctuations.

Scheme dependence of correlation functions (but not of physical amplitudes) is a general feature of renormalization in QFT. When adding counterterms that cancel the cutoff dependent divergences, one is free to add finite counterterms, which are normally chosen such that certain renormalization conditions are satisfied at some renormalization scale. The same holds in HR in AdS/CFT. Also the order-by-order approach presented here includes scheme dependence, although in a more indirect way. The choice of scheme is reflected in certain ambiguities in the choice of the asymptotic basis functions for the fluctuations.

The rest of the review is structured as follows. In Section 2, we review the linearized bulk dynamics of a fake SUGRA system around BPS domain wall backgrounds, expressed in the language of gauge-invariant variables introduced in [31–33], and we introduce the asymptotic decomposition of the on-shell fields into dominant and subdominant basis solutions. The holographic renormalization for two-point functions following the order-by-order approach of [29] is presented in Section 3. We restrict our attention to the scalar sector, but the recipe can be extended easily to the traceless transversal fluctuations of the metric. Because the results of this part yield new insights into the pole structure of the two-point functions, we will also include the relevant updated material from [45]. Then, it will be discussed how scheme dependence is incorporated in the order-by-order approach. Section 4 is dedicated to examples with aAdS backgrounds, that serve to verify the agreement of the results of Section 3 with known results from AdS/CFT. Apart from the pure AdS case, we will consider the GPPZ [56] and the Coulomb branch (CB) flows [57, 58], which have been extensively studied in the literature. The GPPZ flow represents a massive deformation of SYM, while the CB flow is dual to a vacuum of SYM, in which the conformal invariance is spontaneously broken by a VEV. This gives us enough confidence to carry on and consider the KS case in Section 5. Finally, Section 6 contains our conclusions.

2. Bulk Dynamics

Let us start by reviewing the equations governing the dynamics of the bulk fields [32, 33], which encode the information about two-point functions in holographic renormalization group flows.

The systems we consider are fake SUGRAs in dimensions with actions of the form
with , and where the potential is given in terms of a superpotential by
We will not specify at this point the boundary terms in (2.1), as they do not affect the bulk dynamics, although they are important for holographic renormalization. We will come back to them in Section 3. In (2.1) and (2.2), is the metric on the sigma-model target space and is its inverse. Moreover, we used the notation .

Holographic renormalization group flows are described by domain wall backgrounds of the form
with , satisfying the BPS equations
Linearized fluctuations around such a domain wall background are best described in a gauge invariant fashion, which we review next.

2.1. The Sigma-Model Covariant Field Expansion

It is our aim to study the dynamics of the fake supergravity system (2.1), (2.2) on some known backgrounds of the form (2.3), (2.4). In this section, we will expand the fields around the background, exploiting the geometric nature of the physical variables to achieve a gauge-invariant formulation of the fluctuation dynamics.

As is well known in gravity, reparametrization invariance of space-time comes at the price of dragging along redundant metric variables together with the physical degrees of freedom. One usually attempts to reduce redundancy by gauge fixing, but such an approach causes problems for the study of fluctuations in holographic RG flows, due to the coupling between the metric and scalar fluctuations [59–61]. Thus, following [31], we will start from a clean slate keeping all metric degrees of freedom and describe in the next subsection how to isolate the physical ones.

Using the sigma-model metric , one can define the sigma-model connection,
and its curvature tensor,
We also define the covariant field derivative as usual, for example,
All indices after a bar “” are intended as covariant field derivatives according to (2.7). Moreover, field indices are lowered and raised with and , respectively.

Armed with this notation, it is straightforward to expand the scalar fields in a sigma-model covariant fashion. The naive ansatz , introducing simply as the coordinate difference between the points and in field space, leads to noncovariant expressions at quadratic and higher-orders, because these do not form a vector in (tangent) field space. In other words, the coordinate difference is not a geometric object. However, it is well known that a covariant expansion is provided by the exponential map, see for instance [62, 63],
where the higher-order terms have been omitted, and the connection is evaluated at . Geometrically, represents the tangent vector at of the geodesic curve connecting the points and , and its length is equal to the geodesic distance between and ; see Figure 1.

Figure 1: Illustration of the exponential map.

It is also a standard result that the components coincide with the Riemann normal coordinates (RNCs) (with origin at ) of the point (see, e.g., [63]). This fact can be used to simplify the task of writing equations in a manifestly sigma-model covariant form. Namely, given a background point , we can use RNCs to describe some neighborhood of it and then employ the following properties at the origin of the RNC system:
in order to express everything in terms of tensors. Because the background fields depend on , we must be careful to use (2.9) only outside -derivatives, but the simplifications are still significant.

Finally, let us also define a “background-covariant” derivative , which acts on sigma-model tensors as, for example,
If a tensor depends on only implicitly through its background dependence, then we find the identity
The background-covariant derivative will be important in our presentation of the field equations in Section 2.4.

2.2. Gauge Transformations and Invariants

The form of the background solution (2.3) lends itself well to the ADM (or time-slicing) formalism for parametrizing the metric degrees of freedom, compare for example, [62, 64]. Instead of slicing in time, we will write a general bulk metric in the radially-sliced form
where is the induced metric on the hypersurfaces of constant , and and are the lapse function and shift vector, respectively. It will be important to us that the objects , and transform properly under coordinate transformations of the radial-slice hypersurfaces.

We can now expand the radially-sliced metric around the background configuration:
where , and denote small fluctuations. Henceforth, we will adopt the notation that the indices of metric fluctuations, as well as of derivatives , are raised and lowered using the flat (Minkowski/Euclidean) metric, .

Now let us turn to the question of isolating the physical degrees of freedom from the set of fluctuations which we have introduced so far. In the earlier AdS/CFT literature one usually removed the redundancy following from diffeomorphism invariance by partial gauge fixing, that is, by placing conditions on certain components of the metric, such as , . And indeed, it is always possible to perform a change of coordinates which transforms the metric into a form that satisfies the gauge conditions.

Alas, as mentioned above, partial gauge fixing can create problems in coupled systems. Instead, we will obtain the equations of motion in gauge-invariant form. Let us start by considering the effect of diffeomorphisms on the fluctuation fields. We consider a diffeomorphism of the form
where is infinitesimal. Notice that we found it convenient to apply the diffeomorphism inversely, that is, we have expressed the old coordinates in terms of the new coordinates . The use of the exponential map implies that also the transformation laws for the fields can be written covariantly (the functions are thought of as the components of a vector field). For example, a scalar field transforms as
whereas a covariant tensor of rank two transforms as
For the metric tensor , (2.16) is simplified to
Equations (2.15) and (2.16) are most easily derived using RNCs around and exploiting (2.9). The second-order terms in have been included here in order to illustrate the covariance of the transformation laws. For our purposes, the linear terms will suffice.

Splitting the fake supergravity fields into background and fluctuations, as defined in (2.13) and (2.8), the transformations (2.15) and (2.17) become gauge transformations for the fluctuations, to lowest order:
By we mean terms of order in the fluctuations . Moreover, let us decompose as follows (in the following we will always assume ):
where denotes the traceless transverse part, and is a transverse vector. It is straightforward to obtain from (2.18)
The symbol denotes the transverse projector,

The main idea of our approach is to construct gauge-invariant combinations from the fields . Using the transformation laws (2.18) and (2.20), this is straightforward, and to lowest order, one finds the gauge-invariant fields (The choice of gauge-invariant variables is, of course, not unique, as any combination of them will be gauge-invariant as well.)
The variables and both arise from , which has been split into its longitudinal and transverse parts.

Although we have carried out the construction of gauge-invariant variables only to lowest order, and as we will see below, this is all that is needed, it is necessary for consistency that the preceding analysis can, in principle, be extended to higher-orders, which is indeed the case. In this context it becomes clear that the geometric nature of the field expansions, which is implied by the exponential map, is a crucial ingredient of the method.

Finally, let us prepare the ground for the arguments of the next subsection, where we will analyze the implications of gauge-invariance on the equations of motion. To be concise, we continue in a symbolic fashion. Let us consider the set of gauge-invariant fields, . From the definitions (2.22)–(2.26) we see that there is a one-to-one correspondence between and a subset of the fluctuation fields, . We collect also the remaining fluctuation variables into a set, . Henceforth, the symbols , and will be used also to denote the members of the corresponding sets.

One can better understand the correspondence between and by noting that (2.22)–(2.26) can be rewritten as
where is a linear functional of the fields . Going to quadratic order in the fluctuations, one would find
where and are bi-linear in their arguments. Terms of the form do not appear, as they can be absorbed into .

We interpret the gauge-invariant variables as the physical degrees of freedom, whereas the variables represent the redundant metric variables. This can be seen by observing that one can solve the transformation laws (2.20) for the generators , which yields equations of the form
with being a linear functional.

2.3. Einstein's Equations and Gauge Invariance

It is our aim to cast the equations of motion into an explicitly gauge-invariant form. This means that the final equations should contain only the variables and make no reference to and . Reparametrization invariance suggests that this should be possible, and we will establish the precise details in this subsection.

Let us consider Einstein's equations, symbolically written as
but it is clear that the arguments given below hold also for the equations of motion for the scalar fields. To start, let us expand the left-hand side of (2.30) around the background solution, which yields, symbolically,
Here, and denote linear and bilinear terms, respectively. The background equations are satisfied identically. Substituting for using (2.28) yields
Notice that the functionals and are unchanged ( is just replaced by ), whereas the others are modified by the -dependent terms of (2.28), which we indicate by adorning them with a tilde. For example, receives contributions from , and .

In order to simplify (2.32), we consider its transformation under the diffeomorphism (2.14). On the one hand, from the general transformation law of tensors (2.16) we find, using also (2.29), that it should transform as
On the other hand, the variation of (2.32) is
Let us compare (2.33) and (2.34) order by order. The absence of first-order terms on the right-hand side of (2.33) implies that
It can easily be checked that this is indeed the case. Then, substituting into the right-hand side of (2.33) yields
Comparing (2.36) with the second-order terms of (2.34), we obtain
Hence, we find that a simple expansion of Einstein's equations yields gauge-dependent second-order terms, but they contain the (gauge-independent) first order equation, and so can consistently be dropped. Happily, we arrive at the following equation, which involves only :
The argument generalizes recursively to higher-orders. One will find that the gauge-dependent terms of any given order can be consistently dropped, because they contain the equation of motion of lower orders.

Equation (2.38) and its higher-order generalizations are obtained using the following recipe:

Expand the equations of motion to the desired order dropping the fields and replacing every field by its gauge-invariant counterpart .

This rule is summarized by the following substitutions:
Since is traceless and transverse, the calculational simplifications arising from (2.39) are considerable.

Let us conclude with the remark that, although the rules (2.39) can be interpreted as the gauge choice , the equations we found are truly gauge invariant.

2.4. Equations of Motion

In this section, we will put into practice what we have just learned. The equations of motion that follow from the action (2.1) are
for the scalar fields, and Einstein's equations
Notice that we use the opposite sign convention for the curvature with respect to [31, 32].

We are interested in the physical, gauge-invariant content of (2.40) and (2.41) to quadratic order in the fluctuations around an RG flow background of the form (2.3), (2.4). As we have learned in the last section, it is obtained by expanding the fields according to (2.13) and (2.8) and then applying the substitution rules (2.39). Since we defined the expansion (2.8) geometrically, it is assured that we will obtain sigma-model covariant expressions. To carry out this calculation in practice, it is easiest to use RNCs at a given point in field space, so that one can use the relations (2.9) outside -derivatives.

In the following, we will present the linearized equations of motion. We just collect the results without going into details of the derivation. For intermediate steps we refer the reader to the appendices B and C of [33]. (Note that we use a different notation for the indices here than in [33]. The dimensional indices are denoted by here and by there, whereas the dimensional indices are denoted by here and by there.) Let us start with the equation of motion for the scalar fields (2.40), which gives rise to the following fluctuation equation:
Second, the normal component of Einstein's equations gives rise to
Third, the mixed components of (2.41) yield

The appearance of the fields , , and on the left-hand sides of (2.42)–(2.44) seems to indicate the coupling between the fluctuations of active scalars (nonzero ) to those of the metric, which is familiar from the AdS/CFT calculation of two-point functions in the literature. However, the gauge-invariant formalism resolves this issue, because (2.43) and (2.44) can be solved algebraically (in momentum space) for the metric fluctuations , and , so that the coupling of metric and scalar fluctuations at linear order is completely disentangled. One easily obtains (using our assumption )

We proceed by substituting (2.45) and (2.46) into (2.42), using also the identities
which follow from (2.2) and (2.4), and we end up with the second-order differential equation
where we introduced the matrix
Equation (2.49) is the main result of the gauge-invariant approach and governs the dynamics of scalar fluctuations around generic Poincaré-sliced domain wall backgrounds.

Let us also consider the tangential components of (2.41). Because of Bianchi's identity, their trace and divergence are implied by (2.42), (2.43), and (2.44), which is easily checked at linear order. Thus, we can use the traceless transverse projector,
in order to obtain the independent components. This yields
As expected, the physical fluctuations of the metric satisfy the equation of motion of a massless scalar field.

In the following, we focus on the scalar field equation (2.49). Let be the number of scalar fields (components of ). As in [45], we will assume the existence of a set of independent solutions of (2.49), which are defined as power series in (in momentum space), with -dependent coefficients that are more and more suppressed with increasing powers of . (This is tantamount to demanding that the warp function grows without limit for , so that in (2.49) can be regarded as a correction in the asymptotic region.) Moreover, the leading term (for large ) in each solution should be independent of . In position space, simply translates to the operator . Amongst these solutions, one can distinguish between asymptotically dominant solutions () and subdominant solutions with respect to their behaviour at large . Including the field index, we will interpret and as matrices. A regularity condition in the bulk interior, that is,
allows only for independent regular combinations of the asymptotic basis solutions. Hence, we will decompose a general regular solution of (2.49) into
where and are called the source and response coefficients, respectively, and . The bulk regularity condition uniquely determines the (functional) dependence of the responses on the sources and gives rise to the nonlocal information for the two-point functions of the dual operators.

Throughout the paper, we will consider mostly the analogue of (2.54) in momentum space, sometimes omitting the dependence on . Moreover, a will be used to denote the inner product in field space, or the contraction of field space indices, for example, .

3. Perturbative Holographic Renormalization

3.1. Scalar Two-Point Functions

In this section, we will present the general formalism for obtaining finite, renormalized two-point functions for the QFT operators that are dual to the bulk scalar fields. Our starting point is the following action, which is quadratic in the fluctuations and encodes the bulk field equations (2.49),
with some symmetric counterterm matrix , which is a local operator that will be specified in a moment. The bulk integral in (3.1) is to be understood with a cutoff , where also the boundary counterterm is evaluated. It follows that the variation of the on-shell action with respect to a variation of the boundary value is given by
where the right-hand side is evaluated at . Let us define the counterterm matrix as
where is the inverse of the matrix , defined in momentum space as a series in , or equivalently, in position space as a series in . We will see momentarily that this definition leads to finite one- and two-point functions. We also note the following subtlety. The counterterm in (3.1) needs to be local in the fields, which means that should be a polynomial in (in momentum space) or (in position space). The assumptions made in Section 2 imply that is a series in . However, we also assumed that the coefficients of the series with increasing powers of are suppressed for large due to the factor , so that we can truncate the series in (3.3) to some polynomial, because the terms thus neglected vanish in the large- limit. Hence, strictly speaking, the counterterm operator in (3.1) is a polynomial truncation of (3.3).

Before deriving the two-point function, let us also introduce the following matrices:
These matrices are independent of , as one can show from the field equation (2.49). This implies that should be identically zero, as the subdominant solutions vanish fast enough asymptotically. (This is not necessarily the case if there are two or more bulk scalars with mass , which, in the aAdS-case, would be dual to operators of dimension . If at least two of these scalar fields are present and the background is not aAdS, one has to check more carefully whether indeed vanishes. We will assume this in the following, as it simplifies our final result. In all the examples we are considering later, this issue does not play any role.) Furthermore, they are functions of or , depending on whether one works in momentum or position space.

Combining (3.2) with the decomposition (2.54), the (linear term of) the exact one-point function (1.2), in momentum space, takes the form (The subscript on the left-hand side indicates that these are just the terms linear in the fluctuations.)
where, for the sake of brevity, we have omitted the dependence of the asymptotic solutions and on and .

Substituting (3.3) into (3.5) and using the matrices (3.4), after some algebra one obtains
Here, we have omitted the arguments on the right-hand side. To obtain the final result, we observe that the third term on the right-hand side vanishes, since , as stated above. Moreover, the last term, which is the only one with a cutoff dependence, vanishes when the large- limit is taken, because goes to zero. Hence, we end up with
which holds both in momentum and position space. From (3.7), one obtains the connected two-point function
As promised, (3.7) and (3.8) are finite in the limit , as the matrices and do not depend on . Equation (3.7) agrees with of [45], for which it provides the missing piece and identifies the contact term, which will be discussed further in Section 3.3.

In momentum space, (3.8) has a more practical form. Setting by translational invariance and Fourier transforming the coordinate , one finds
In what follows, we will often work in momentum space omitting the argument . By the two-point function in momentum space, we will intend (3.9).

Unfortunately, the symmetry of the two-point function under exchange of and is not obvious from (3.8). It would be a nontrivial test of any concrete calculation to see whether the right-hand side, with its antisymmetric second term, combines to something symmetric.

A different form of the 2-point function, which makes its pole structure more explicit, can be obtained with the help of the Green's function. The Green's function satisfies
where the factor on the right-hand side is the metric factor from the covariant delta function. The Green's function can be written in terms of a basis of eigenfunctions,
where the functions satisfy (2.49) for . (Again, we omit the matrix indices, and the indices of the two 's are not contracted.) Substituting (3.11) into (3.10) yields the completeness relation
from which one can deduce the orthogonality relation
With the dot product we denote the covariant contraction of indices. Equation (3.13) provides the condition for the eigenstates to be integrable. Due to the factor , the integral measure is not the covariant bulk integral measure that one might have expected.

It can be checked in the various cases we consider that the asymptotically dominant behaviours, , are not integrable, whereas the subdominant behaviours are integrable. Thus, we have

To derive a form of the 2-point function which makes its symmetry and pole structure manifest, we start by considering the general formula for a solution of (2.49) in terms of the Green's function and prescribed boundary values. Let be a (large) cutoff parameter determining the hypersurface where the boundary values are formally prescribed. Remembering that neither the Green's function nor its derivative vanish at the cutoff boundary, we have (This formula follows from (3.10) upon multiplication by from the left, taking the integral over , integrating by parts and using the field equation (2.49). The IR boundary does not contribute, because vanishes there. The reason for this is that should correspond to a single point of the bulk space, which is only guaranteed if vanishes there, c.f. (2.3).)
where and are the prescribed values of the field and its first derivative at the cutoff boundary, respectively. Since is an unphysical cutoff parameter, we must ensure that the bulk field remains unchanged when is varied. This is easily achieved, if, together with a change of the cutoff, , the boundary values are changed by
and the second derivative, , is determined by the equation of motion (2.49). To assure (3.16), we determine the formal boundary values at the cutoff, and , from the generic asymptotic behaviour (2.54) (in momentum space), with coefficients and fixed. After inserting (2.54) and (3.11) into (3.15), we obtain

To continue, we observe that for very large , the term on the second line of (3.17), containing only subdominant solutions, is much smaller than the term on the third line. Therefore, we drop it. Moreover, as we are interested only in the pole structure, we consider very close to and expand the numerator keeping only the leading term, that is, we replace by in the numerator. Finally, we use the fact that the eigenfunctions are purely subdominant, (At this point one may wonder where the dominant part of comes from. It arises from the sum over the spectrum in (3.17), in particular from the UV contribution. For the simple case of AdS bulk, this is shown in appendix A.1 of [45]. However, it does not contribute to the poles.)
This yields
with defined in (3.4).

Thus, after reading off the response function (for ) from (3.19), we obtain the poles of the connected 2-point function, using (3.7) and differentiating with respect to the source ,
Therefore, defining also
our final result is
We note that the are independent of the choice of the subdominant basis solutions, because the normalization of the eigenfunctions is fixed by (3.13). Moreover, (3.22) shows that the pole terms of the 2-point functions are manifestly symmetric under exchange of and .

3.2. VEVs

Let us make a few comments on VEVs. Equation (3.5) only gives the part of the one-point function which is linear in the fluctuations. At the moment, our approach does not allow for a systematic derivation of the VEVs yet. However, we would like to make the following observation. The scalar equations (2.49) have the zero mode solution ()
which only depends on the radial variable . Like any fluctuation, this has an expansion into dominant and subdominant asymptotic solutions as in (2.54). In aAdS settings, nonzero coefficients of the dominant and subdominant basis solutions are interpreted as finite couplings and VEVs, respectively. As mentioned in the introduction, the interpretation of finite couplings (in the sense of a deformation of some special QFT) cannot be made in the general case, but we will continue to interpret the presence of asymptotically subdominant behaviour as finite VEVs. However, this just means that the VEV is nonzero, because the normalization factor in (3.23) is undetermined. In addition, there is an issue of scheme dependence, when nonzero dominant behaviours are present, but this happens also in aAdS settings, as will be seen in Sections 4 and 5.

3.3. Scheme Dependence

In QFT, contact terms of correlation functions, which do not influence physical scattering amplitudes, depend on the renormalization scheme. Let us now discuss how the scheme dependence of the two-point functions (3.8) appears from a bulk point of view. For the sake of brevity, we will work in momentum space and omit all functional arguments.

The starting point is the decomposition (2.54) of a regular solution to the bulk field equations. Clearly, the definition of the asymptotic solutions and is not unique. Our restriction on the functional form of these solutions in terms of series of and the fact that all subdominant solutions are negligible for large with respect to all dominant ones still allows for a change of basis of the form
where the (nondegenerate) matrices , and are polynomials in . Under this change of basis, the matrices and transform into (Remember .)
respectively, while the source and response coefficients in the new basis become
Inserting these transformations into (3.8), one finds the connected two-point functions of the operators coupling to the sources ,
Hence, the matrix performs a rotation of the basis of operators, as one would have expected, while a nonzero changes the contact terms, which corresponds to a change of renormalization scheme.

In QFT, operators are usually characterized by their scaling dimension, which is renormalization scale dependent. Under renormalization, they undergo operator mixing, such that an operator of a given dimension, defined at a certain renormalization scale, is generically made up of the operators of equal and lower dimensions, defined at a larger renormalization scale. There is, however, some ambiguity, as operators of equal dimension and otherwise equal quantum numbers can be arbitrarily combined to equivalent combinations. This ambiguity finds a natural counterpart in the present approach. Ordering the dominant asymptotic solutions according to their asymptotic behaviour in descending order, it is natural to choose in “upper triangular” form, such that each dominant solution is modified only by solutions of equal and weaker asymptotic behaviour. A similar remark would apply for .

A further restriction on the redefinition could come from the fact that the lowest order terms in a near boundary expansion of the dominant solutions typically have a definite correlation between powers of and powers of . This is well known, for instance, in the aAdS case with a single scalar field, compare for example, the discussion in Section of [12]. Something similar happens in the case of KS, as can be seen from the explicit form of the asymptotic solutions given in the appendix of [29] (we refrain from reproducing them here as they are very “bulky”). We will refer to a choice of dominant solutions respecting this correlation as a “natural” choice.

Finally, we remark that it is reasonable to assume that and/or can be chosen such that in (3.25). A possible obstruction to this possibility would be that the matrices needed to achieve that are nonpolynomial in . We will see later that the choice is possible for the KS system. Starting with such a choice, a further change of basis using just would lead to
implying that one can achieve by a suitable choice of , although this choice is obviously not unique.

4. Examples

In this section, we will compare the general expressions from the previous sections with the results of holographic renormalization in the case of aAdS bulk space-times. To carry out HR, one can use one's favourite method from the choice of [10, 11, 14]. We will start with pure AdS and then consider some favourite RG flows, the GPPZ [56] and the Coulomb branch flows [57, 58].

4.1. Scalars in AdS Background

As the simplest case, we consider a number of free massive scalars in a pure AdS background. The results are, of course, dictated by conformal invariance of the boundary CFT, but it is still useful to deal with this case, because several statements that we make in what follows hold in any aAdS configuration. In this and the following subsections we will set the AdS length scale to . It can be reinstated by dimensional analysis.

An AdS background exists for any superpotential with a fixed point (where is nonvanishing). We consider the scalars parametrized by Riemann normal coordinates around the fixed point and their local coordinate system rotated in such a way that the matrix of second derivatives of is diagonal [33]. This gives rise to a set of canonical scalar fields with a superpotential
The -dimensional AdS background metric is given by
where the radial variable is related to of Section 2 by . To linear order, the scalar fields satisfy the equations of motion
where no summation over is intended. The masses are related to the coefficients of (4.1) by

Z matricesFor , the conventionally normalized asymptotic solutions of (4.3) are
(no summation over ) where the are modified Bessel functions, and
The powers of in front of the solutions are necessary in order to make the leading terms -independent. It is straightforward to verify that the matrices (3.4) are (as the solutions are diagonal, that is, proportional to , we use and interchangeably)
These equalities hold true in any aAds configuration, as long as one uses the same asymptotic normalizations.

SpectrumEquation (4.3) admits a continuous spectrum of regular and subdominant solutions for , . The eigenfunctions are given by (The generic label for the eigenfunctions used in Section 3 [c.f. (3.11)] is replaced here by two indices, and . As before, the upper index is the vector component index.)
where the are Bessel functions, and they satisfy the orthogonality relation
Considering the small- behaviour of (4.9) and comparing with (4.6), one can read off the response coefficients of the eigenfunctions, compare for example, (3.18),
Hence, after using (3.21), one obtains the two-point function in the form (3.22) as the sum (here it is an integral) over the spectrum
Notice that the second equality holds only after an analytic continuation, because the integral does not exist if . This is equivalent to adding an infinite contact term to the integral over the spectrum. For example, if , we rewrite the integrand as
and add a contact term that cancels the first term on the right-hand side.

4.2. GPPZ Flow

In the GPPZ flow [56], we consider two canonical scalar fields and with the superpotential
Both are dual to operators with bare dimensions . The scalar is the active scalar of the GPPZ flow, that is, it is nontrivial in the background. More relations for the GPPZ flow can be found in [65, 66].

Counterterms and Scheme DependenceLet us start by reviewing the treatment in standard AdS/CFT [10, 11, 14]. Being dual to operators of dimension 3, the scalar fields behave asymptotically as
where , and are independent coefficients, whereas and depend on and . The background solution satisfies
which implies

Carrying out holographic renormalization [10, 11, 14], one finds the coefficients and ,
as well as the exact one-point functions with the full scheme dependence, (We have omitted the curvature-dependent terms, which are irrelevant here.)
The scheme-dependent coefficients that appear here stem from the addition of finite counterterms of the form

Let us linearize around the background (4.16), (4.17), which is sufficient to extract the information for the two-point functions, and translate the fluctuation into the gauge-invariant variables. One must take some care for the active scalar. Its mixing with the metric fluctuations is described by (2.22), (2.23), and (2.45), where we must set , because of the orthogonal gauge used to derive (4.19). Hence, using
one arrives at
The coefficients , with superscripts and are the independent coefficients of the asymptotic expansion, that is, the analogue of (4.15), of the components of . The first term on the right-hand side of (4.22) is a scheme-dependent, that is, unphysical, VEV, which vanishes in renormalization schemes that respect SUSY () [12].

Let us now compare these expressions with the results of Section 3.1. In particular, let us define the dominant and subdominant solutions by
with two as yet undetermined coefficients and . The second terms in the dominant solution can be determined directly from (4.18). From (4.23), one obtains the counterterm matrix (3.3)
where the ellipses indicate terms which do not contribute and can be truncated. It can be noted that the resulting divergent counterterms agree with the divergent counterterms from the standard approach. The last terms shown give finite contributions.

It is straightforward to verify that the matrices (3.4) that one obtains from (4.23) agree with the AdS result (4.8).

Writing the bulk field in terms of the asymptotic solutions (4.23) gives the source and response coefficients, (the expressions for and are identical)
so that the linear term of the one-point function (3.7) becomes
Comparing this with the linear terms in (4.22), we find agreement in the nonlocal term containing . Moreover, we can determine the constants and as (An alternative way of determining them is to compare the finite counterterms that result from (4.24) with the linearized standard ones.)
This result states explicitly the relation between the choice of the dominant basis and the renormalization scheme. Note that only for or appear.

Spectrum of the Active ScalarLet us illustrate formula (3.22), which represents the 2-point function as a sum over its poles, up to local terms, using as an example the active GPPZ scalar. The gauge invariant equation of motion reads [31, 66]
where the radial coordinate is defined by , and the warp factor is .

Equation (4.28) admits a discrete spectrum of regular and subdominant eigenfunctions, with mass squares
The normalized eigenfunctions are
where are Legendre polynomials. (As in [31, 66], regular and subdominant solutions of (4.28) are given by Jacobi polynomials , which are proportional to .) One easily finds the response coefficients
Thus, we obtain for the 2-point function (3.22),
Clearly, the sum in (4.32) does not converge, so that there are again infinite contact terms. It is instructive to compare (4.32) with the finite result from holographic renormalization [61, 67]. Let us pick the particular SUSY scheme , . Then
where . Using the identity
we obtain from (4.33)
The nonlocal part agrees precisely with (4.32), and the scheme-dependent terms in (4.22) have the same form as the infinite contact terms.

4.3. Coulomb Branch Flow

Let us consider the Coulomb branch (CB) flow [57, 58]. There is a canonical bulk scalar with the superpotential
The background solution is given by the relations
introducing the radial variable . The length is independent of the AdS radius , together with which it determines the radius of the disc on which the D3 branes are distributed ().

Counterterms and Scheme DependenceLet us again briefly review the asymptotic analysis form standard AdS/CFT. It follows from (4.36) that is dual to an operator of bare dimension . Correspondingly, it has an asymptotic expansion of the special form
where and are the two independent coefficients. Asymptotically, the background vanishes at the rate
implying

From HR, the exact one-point function of the corresponding operator is given by [10, 11, 67]
Two comments are in order here. The second term, involving a scheme-dependent constant , is new compared to the corresponding formulas of [10, 11, 67] and arises from the addition of a finite counterterm proportional to
Furthermore, our result differs from [10, 67] by a factor of , as our differs by a factor from theirs and our definition of the one-point function (1.2) exhibits an additional minus sign.

In contrast to the GPPZ flow, there is a scheme-independent VEV,
The presence of this VEV (but not its value) can also be inferred from the background mode,

Linearizing around the background and switching to the gauge invariant scalar, with an asymptotic expansion
the one-point function (4.41) reads

In order to make contact to Section 3.1, let us define the dominant and subdominant solutions as
Writing (4.45) in this basis, one can read off the source and response as
In this case, (3.4) results in and . (The case would imply . The generic (4.8) does not apply, because of the logarithm in the dominant solution.) Thus, (3.7) reads
Comparison with the part of (4.46) that is linear in the fluctuation implies .

Finally, the counterterm “matrix” (3.3) obtained from the basis (4.47) is
This provides the standard logarithmically divergent counterterm and a scheme-dependent finite contribution.

Two-Point Function and SpectrumThe two-point function for the CB flow was calculated in [61, 67]. Here, we present the calculation using the equation of motion in the gauge-invariant formalism. To obtain the two-point function of the scalar operator dual to , consider the equation of motion (2.49), which can be written in the form
where . For convenience, we have set also . The regular solution of (4.51), up to an irrelevant normalization, is found to be